A White Noise Approach to Stochastic Differential Equations Driven by Wiener and Poisson Processes

Abstract

The theory of Gaussian white noise and its applications to stochastic differential equations (SDEs), both ordinary and partial, are now well-known. The Poissonian white noise theory has similar applications to SDEs driven by the Poisson process. This chapter briefly reviews the white noise theory in the Gaussian and Poissonian cases separately and some relations between them. It outlines how to combine the two white noise theories into one theory that cover equations driven by both Gaussian and Poissonian noise. The chapter illustrates the theory by applying it to some stochastic ordinary and partial differential equations driven by a Brownian motion and an independent Poisson process.